Problem 17 - Thomas Calculus

Question

Evaluate the integrals in Exercises \(1-28\). $$ \int \frac{8 d w}{w^{2} \sqrt{4-w^{2}}} $$

Step-by-Step Solution

Answer

The integral evaluates to \(-4\frac{\sqrt{4 - w^2}}{w} + C\)."

Step 1: Simplify the Integrand

First, observe that the integral is given by \( \int \frac{8 \, dw}{w^{2} \sqrt{4-w^{2}}} \). Notice that the expression under the square root, \( \sqrt{4-w^2} \), suggests a trigonometric substitution because it resembles the form \( \sqrt{a^2 - x^2} \) where \( a = 2 \). This calls for the substitution \( w = 2 \sin \theta \).

Step 2: Substitute and Compute dθ

Using the substitution \( w = 2 \sin \theta \), we have \( dw = 2 \cos \theta \, d\theta \). The integral then becomes: \[ \int \frac{8 \, (2 \cos \theta) \, d\theta}{(2 \sin \theta)^2 \sqrt{4-(2 \sin \theta)^2}}. \] Simplifying \( \sqrt{4-(2 \sin \theta)^2} = \sqrt{4 - 4 \sin^2 \theta} = \sqrt{4 \cos^2 \theta} = 2 \cos \theta \).

Step 3: Simplified Integral

Substitute \( 2 \cos \theta \) in the integral: \[ \int \frac{16 \cos \theta \, d\theta}{4 \sin^2 \theta \, (2 \cos \theta)} = \int \frac{16 \, d\theta}{4 \sin^2 \theta}. \] Simplify it to: \[ \int \frac{4 \, d\theta}{\sin^2 \theta}. \] This can be rewritten as \( 4 \int \csc^2 \theta \, d\theta \).

Step 4: Integrate with Respect to θ

The integral \( \int \csc^2 \theta \, d\theta \) is a standard integral, which equals \( -\cot \theta \). Therefore, \( 4 \int \csc^2 \theta \, d\theta = -4 \cot \theta + C \), where \( C \) is the constant of integration.

Step 5: Back-Substitute for w

Re-substitute the original variable \( w \). We used \( w = 2 \sin \theta \), which implies \( \sin \theta = \frac{w}{2} \). Thus, \( \cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{\sqrt{1 - \sin^2 \theta}}{\sin \theta} = \frac{\sqrt{1 - (\frac{w}{2})^2}}{\frac{w}{2}} = \frac{\sqrt{4 - w^2}}{w} \). So, our solution becomes \( -4 \frac{\sqrt{4 - w^2}}{w} + C \).

Key Concepts

Integrals

Integrals are a fundamental concept in calculus, often described as the reverse process of differentiation. They represent the accumulation of quantities, such as areas under a curve, or general anti-derivatives. Understanding how to solve integrals is crucial in various fields of science and engineering. There are two main types of integrals:

• Definite integrals: These calculate the net area under a curve between two points. They're often used to determine quantities like total displacement or total area.
• Indefinite integrals: These find a general formula for the anti-derivative of a function. The result includes an arbitrary constant, usually written as + C, to account for the infinite possible anti-derivatives.

The given problem involves finding an indefinite integral. The presence of a square root in the denominator and a power of a variable suggests the use of a particular technique to simplify the process. Identifying patterns in the integral helps to choose the best method for solving it, in this case, using trigonometric substitution.

Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between the angles and sides of triangles. It's essential for solving problems involving right-angled triangles and is widely used in physics, engineering, and navigation. Some basic trigonometric functions include sine (\(\sin \theta\)), cosine (\(\cos \theta\)), and tangent (\(\tan \theta\)). Each of these functions relates an angle to a ratio of two sides of a right triangle.In the context of the integral problem, we notice the expression \( \sqrt{4 - w^2} \). This form is suggestive of the trigonometric identity \(\sqrt{a^2 - x^2}\), which often prompts the use of substitution involving sine or cosine. For our problem, substituting \(w = 2 \sin \theta\) uses the Pythagorean identity \(1 - \sin^2 \theta = \cos^2 \theta\), simplifying the integration process. Understanding these identities and substitutions is key for tackling integrals involving \(\sqrt{a^2 - x^2}\).

Substitution Method

The substitution method is a powerful technique in calculus used to simplify complicated integrals, especially when they involve composite functions or nested expressions. By introducing a new variable, we can transform the integrand into a simpler form, making it easier to integrate.In our exercise, we applied trigonometric substitution, a specific type of substitution method. Here's the process:

• Identify the substitution: Based on the \(\sqrt{4-w^2}\) form, we decide on \(w = 2 \sin \theta\) to leverage trigonometric identities.
• Transform the differential: Calculate \(dw = 2 \cos \theta \, d\theta\) to change the variable of integration.
• Simplify the integral: Substitute \(w\) and \(dw\) into the original integral, simplifying terms like \(\sqrt{4 - w^2} = 2 \cos \theta\).
• Solve and back-substitute: Perform the integration using known integrals and trigonometric identities, then return to the original variable.

This method, especially with trigonometric substitution, is invaluable for handling integrals that seem complex at first glance but fit a recognizable pattern when trigonometric identities are applied.